Factorization system

In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.

A factorization system (E, M) for a categoryC consists of two classes of morphismsE and M of C such that:

1. E and M both contain all isomorphisms of C and are closed under composition.
2. Every morphism f of C can be factored as ${\displaystyle f=m\circ e}$ for some morphisms ${\displaystyle e\in E}$ and ${\displaystyle m\in M}$.
3. The factorization is functorial: if ${\displaystyle u}$ and ${\displaystyle v}$ are two morphisms such that ${\displaystyle vme=m'e'u}$ for some morphisms ${\displaystyle e,e'\in E}$ and ${\displaystyle m,m'\in M}$, then there exists a unique morphism ${\displaystyle w}$ making the following diagram commute:

Remark:${\displaystyle (u,v)}$ is a morphism from ${\displaystyle me}$ to ${\displaystyle m'e'}$ in the arrow category.

Two morphisms ${\displaystyle e}$ and ${\displaystyle m}$ are said to be orthogonal, denoted ${\displaystyle e\downarrow m}$, if for every pair of morphisms ${\displaystyle u}$ and ${\displaystyle v}$ such that ${\displaystyle ve=mu}$ there is a unique morphism ${\displaystyle w}$ such that the diagram

commutes. This notion can be extended to define the orthogonals of sets of morphisms by

${\displaystyle H^{\uparrow }=\{e\quad |\quad \forall h\in H,e\downarrow h\}}$ and ${\displaystyle H^{\downarrow }=\{m\quad |\quad \forall h\in H,h\downarrow m\}.}$

Since in a factorization system ${\displaystyle E\cap M}$ contains all the isomorphisms, the condition (3) of the definition is equivalent to

(3') ${\displaystyle E\subseteq M^{\uparrow }}$ and ${\displaystyle M\subseteq E^{\downarrow }.}$

Proof: In the previous diagram (3), take ${\displaystyle m:=id,\ e':=id}$ (identity on the appropriate object) and ${\displaystyle m':=m}$.

The pair ${\displaystyle (E,M)}$ of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions:

1. Every morphism f of C can be factored as ${\displaystyle f=m\circ e}$ with ${\displaystyle e\in E}$ and ${\displaystyle m\in M.}$
2. ${\displaystyle E=M^{\uparrow }}$ and ${\displaystyle M=E^{\downarrow }.}$

Suppose e and m are two morphisms in a category C. Then e has the left lifting property with respect to m (respectively m has the right lifting property with respect to e) when for every pair of morphisms u and v such that ve = mu there is a morphism w such that the following diagram commutes. The difference with orthogonality is that w is not necessarily unique.

Un sistema de factorización débil ( E , M ) para una categoría C consta de dos clases de morfismos E y M de C tales que: ^{[ 1 ]}

1. La clase E es precisamente la clase de morfismos que tienen la propiedad de levantamiento izquierdo con respecto a cada morfismo en M.
2. La clase M es precisamente la clase de morfismos que tienen la propiedad de levantamiento derecha con respecto a cada morfismo en E.
3. Cada morfismo f de C puede factorizarse como${\displaystyle f=m\circ e}$para algunos morfismos${\displaystyle e\in E}$y${\displaystyle m\in M}$.

Esta noción conduce a una definición sucinta de categorías modelo : una categoría modelo es un par que consta de una categoría C y clases de (las llamadas) equivalencias débiles W , fibraciones F y cofibraciones C de modo que

• C tiene todos los límites y colímites,

• ${\displaystyle (C\cap W,F)}$es un sistema de factorización débil,

• ${\displaystyle (C,F\cap W)}$es un sistema de factorización débil, y

• ${\displaystyle W}$satisface la propiedad de dos de tres: si${\displaystyle f}$y${\displaystyle g}$son morfismos componibles y dos de${\displaystyle f,g,g\circ f}$están en${\displaystyle W}$, entonces también lo es el tercero. ^{[ 2 ]}

Una categoría modelo es una categoría completa y cocomplete equipada con una estructura modelo. Un mapa se llama fibración trivial si pertenece a${\displaystyle F\cap W,}$y se denomina cofibración trivial si pertenece a${\displaystyle C\cap W.}$Un objeto${\displaystyle X}$se llama fibrante si el morfismo${\displaystyle X\rightarrow 1}$al objeto terminal es una fibración, y se llama cofibrante si el morfismo${\displaystyle 0\rightarrow X}$a partir del objeto inicial es una cofibración. ^{[ 3 ]}

• Riehl, Emily (2008), Sistemas de factorización (PDF)